SCHWARTZ, LAURENT

Schwartz is the inventor of distribution theory, now a universally used language of mathematical analysis. He also introduced the concept of radonifying maps, related to both the geometry of Banach spaces and probability theory.

Life and Career Laurent Schwartz was the first son of Anselme Schwartz, originally an immigrant from Alsatia, at the time under German control, and later a highly successful surgeon. (In 1907 he became the first Jewish surgeon ever officially employed in a Paris hospital.) Through his mother, Claire, Laurent was related to the Debrés, a prominent Jewish French family: his maternal grandfather was the chief rabbi in Neuilly; later, a Debré would be president of the national Academy of Medicine; there have been and still are prominent Gaullist politicians in the Debré family. (These had become Catholic converts.) In 1938 Laurent Schwartz married Marie-Hélène Levy, whose father, Paul Levy, is the initiator of modern probability theory; Marie-Hélène Schwartz was to become a distinguished mathematician in her own right. The eminent mathematician Jacques Hadamard was Laurent’s granduncle.

On completing the lycée (high school), Laurent Schwartz won, in the Latin category, the Concours Général, the most prestigious nationwide competition in France for high schoolers. But being attracted to geometry as much as to the classics, he applied to and was admitted into the science classes of the École normale supérieure (ENS) of Paris, the most selective and most scholarly oriented of the Grandes écoles, the elite specialized colleges parallel to the universities in the French system of higher education. The ENS students had the privilege of attending the lectures of some the best French mathematicians: Émile Borel, Élie Cartan, Alfred Denjoy, Maurice Fréchet, George Julia, and Paul Montel. At the nearby Collège de France in Paris, they could also hear Henri Lebesgue’s lectures and take part in the Hadamard seminar. The enduring love of Laurent Schwartz for probability theory developed through his personal contact with his future father-in-law, Paul Lévy.

In 1937, having completed his studies at the ENS and having secured the aggrégation, the diploma required for a teaching position in a lycée, Schwartz was drafted into the military, a three-year term being compulsory in France at that time. War came in 1939, defeat in 1940, and the Schwartzes were forced to move to the south of France to avoid German occupation. For a while and despite the fact that he was Jewish, Laurent Schwartz was paid a stipend by the organization that was to become,
after the war, the Centre national de la recherche scientifique (National Center for Scientific Research). After the end of his employment there in 1942, his research was supported until 1944 by the Michelin foundation ARS; he thus circumvented the racial policies of the Vichy government.

A chance encounter with Henri Cartan and Jean Delsarte in Toulouse in 1941 encouraged the Schwartzes to move to the University of Clermont-Ferrand. There they both could work under the guidance of mathematicians who had migrated from the occupied northern zone: Jean Dieudonné, Charles Ehresman, André Lichnerowicz, and Szolem Mandelbrojt, among others. At Clermont-Fer-rand, Laurent Schwartz completed his PhD thesis on the approximation of continuous functions on R+=[0,+∞) by sums of exponentials S=a0+Σnaiexp(–λn> x where the infinite sequence of real numbers λn> 0 is fixed. A classical theorem of Charles Müntz implies that these sums are dense in C(R+) if and only if μnλn-1=+∞. Schwartz proved that when μnλn-1=+∞. the closure in C(R+) of the subspace made up of the sums S consists of the functions that can be extended holomorphically to the open half-plane Rz > 0.

In Clermont-Ferrand he met and was deeply influenced by several members of the Bourbaki group. Nicholas Bourbaki, the fictitious author of foundational mathematical treatises, was the creation of a small number of French mathematicians (Henri Cartan, Claude Chevalley, Jean Dieudonné, Jean Delsarte, René de Possel, and André Weil) determined to realize the axiomatization of mathematics as envisaged earlier by David Hilbert. The membership in the entity Bourbaki was ever changing, upcoming young mathematicians replacing departing older ones. Schwartz was soon recruited into the elite group.

In November 1944, after the Allies had liberated most of France, Laurent Schwartz made the paramount discovery of his life, distribution theory. After teaching at the University of Grenoble during the academic year 1944–1945, he joined the faculty in Nancy. In 1950 he was awarded the Fields Medal for his discovery of distributions. In 1952 he was named to a professorship at the Sorbonne. Several of Schwartz’s PhD students at the Sorbonne have had mathematical careers of great distinction, among them Louis Boutet de Monvel, Alexandre Grothendieck, Jacques-Louis Lions, Bernard Malgrange, and André Martineau. In 1969 he moved to the École polytechnique (Polytechnic School) in Paris (later moved to Palaiseau), the top engineering school in France, where Paul Lévy had taught and where Schwartz embarked on an ambitious, and successful, program of reform. At Polytechnique he organized a very productive seminar on infinite dimensional measure theory, attracting a number of brilliant students, such as Bernard Maurey and Gilles Pisier (whose PhD thesis he supervised).

During his entire adult life, Schwartz was active in left-wing politics, a follower of Trotsky as a young man before abandoning the movement after World War II, then agitating against France’s Algerian war and France’s and later America’s war in Vietnam. He used his scientific prestige to combat Abūses of human rights, especially (but not solely) those of mathematicians and scientists in South America and in the Soviet states.

Schwartz was also well-known among lepidopterists. His collection of tropical butterflies and moths, one of the greatest private collections in the world (with more than twenty-five thousand specimens), was donated in part to the Muséum d’histoire naturelle in Paris, in part to the natural history museum in Cochabamba, Bolivia. His collecting throughout tropical Latin America led to the discovery and description of several new species, now named after him.

Distribution Theory The start of the twentieth century saw a number of attempts to define derivatives of functions not commonly thought of as differentiable (e.g., step functions), mainly prompted by the need to solve ordinary differential equations (ODEs) or partial differential equations (PDEs). Of such attempts, the two most significant were the symbolic calculus of the British physicist Oliver Heaviside, devised to solve the ODEs of electrical circuits, and Hadamard’s finite parts, introduced to make sense of the fundamental solutions (in modern parlance) of the wave equation in higher space-dimensions. Heaviside’s calculus transformed convolution into multiplication, but at the price of introducing derivative rules for the convolution
of two differentiable functions in the closed half-line [0,+∞),

that were hard to understand. Mathematicians who came after Heaviside gave a definition of Heaviside’s calculus in terms of the Laplace transform, which, however, was not very convincing. Meanwhile, the first generation of quantum physicists was producing more examples of generalized functions with an impressive array of applications. Among them was the Dirac delta function δ (x)(known earlier to Heaviside): it was the function with symbol 1, and it was presented as the derivative of what was later called the unit step or Heaviside function, which is equal to zero in (-∞, 0) and to one in (0,+∞), whose symbol is 1/p. Paul Dirac did not explicitly connect his intuitive but less than rigorous ideas to Heaviside’s calculus, but instead by approximating, defined his “function” with true functions. In the course of this work he introduced
multidimensional Dirac measures, for instance, the measure associated to the light-cone Γ R4,

where dμ is the natural invariant measure on Γ in (2) can be any continuous function in R4 decaying rapidly at infinity.

Further developments of this highly effective but poorly explained mathematics happened in the late 1920s and the 1930s. In 1926 Norbert Wiener used regularization or smoothing, that is, convolution with compactly supported C∞ functions, to approximate continuous functions f by smooth ones. Salomon Bochner defined the Fourier transform of finite sums of the form

in which Pn is a polynomial and fn;L2(Rn) (NZ+ can vary). This is essentially the definition of what in Schwartz’s later theory was to be called a tempered distribution, but the differences are nonetheless considerable. Bochner considered the derivatives dn/dxn purely as formal operations, not as weak derivatives, and although the space of sums (3) is stable by (formal) Fourier transform, which exchanges multiplication and convolution, Bochner did not specify when these operations are defined. Nor did he mention that the Dirac function is of the type (3). Indeed, when in 1946 Bochner defined the generalized (or, as would be said today, the distributional) solution of a linear ODE, he went no further in that direction.

In 1934–1935, Jean Leray gave the prestigious Peccot lectures at the Collège de France, which Schwartz attended. In them, Leray defined the weak solution u of a second-order linear PDE in R3, P(x, x) u = 0, by the property that

for every compactly supported C2 function; here PT(x,x)) denotes the transpose of P(x,x)). The coefficients of P(x,x)) must have some regularity, but u is only required to be locally integrable.

The Soviet mathematician Sergei L’vovich Sobolev came the closest of anyone in the 1930s to the discovering the general concept of a distribution. In his articles “Méthode nouvelle à résoudre le problème de Cauchy” (1936) and “Sur un théorème d’analyse fonctionnelle” (1938), he defined the distributions of a given, arbitrary,

finite order m as the continuous linear functionals on the space Cmcomp of compactly supported functions of class Cm. Using transposition, Sobolev defined the multiplication of the functionals belonging to by the functions belonging to (Cmcomp)and observed that the differentiation of those functionals maps (Cmcomp)* into (Cmcomp)*. But he made no mention of Dirac’s δ (x), and neither convolution nor the Fourier transform play a role in his theory. At this time he limited himself to applying his new approach to reformulating and solving the Cauchy problem for linear hyperbolic equations. Only after the war did he introduce the Sobolev spaces Hm, and then only for integers m≥ 0. World War II and the separation between Western and Soviet mathematicians left Schwartz ignorant of Sobolev’s papers. No doubt knowing them would have greatly facilitated the final construction of distribution theory. Sobolev also kept the integer m fixed; he never considered the intersection C∞Compof the spacesCmcompfor all m, which is surprising since he proved that is C∞+1Comp dense in Cmcompby Wiener’s smoothing method.

In his formulation, Laurent Schwartz used the language and tools of functional analysis, essentially those of
the landmark monograph of Stefan Banach, Théorie des opérations linéaires (1932): a distribution is a linear functional on the space Ccompof test functions in an open subset of Euclidean space or, more generally, in a smooth manifold. The functional must be continuous, in the sense that its value on a sequence of test functions, whose derivatives of all orders converge uniformly and whose supports remain confined to a fixed compact subset, must converge. The basic operations of analysis, differentiation, and multiplication by smooth functions, are defined in the space D' of all distributions by transposition (thus derivative=weak derivative); and so is the convolution of two distributions (under reasonable hypotheses on their decay at infinity). The scale of Sobolev spaces Hs and their variants are seamlessly integrated in the array of distribution spaces (spaces topologically embedded into D'). On Euclidean space Rn the interpretation of Hs via Fourier transform is practically self-evident.

A major contribution of the theory was the selection of the Schwartz space S of rapidly decaying C∞ functions at infinity and of its dual S’, the space of tempered distributions (see above), as the proper framework for Fourier analysis (ideas missing from Bochner’s earlier analysis). This choice was not preordained: there are many other spaces stable under Fourier transform, spaces of functions that decay much faster at infinity. But other choices would run contrary to the deeper underlying uncertainty principle: the temperedness of tempered distributions ensures the localizability of their Fourier transform (think of analytic functionals whose Fourier transform can grow exponentially and that are not here nor there). Certain spaces of Gevrey ultradistributions are also localizabile, but they are of incomparably narrower usefulness.

Another remarkable feature of distribution theory is the Schwartz kernel theorem. It states a fundamental property of the main distribution spaces: In certain respects they are more like finite-dimensional Euclidean space than infinite-dimensional Banach spaces. Specifically, in any one of them, C∞, C∞Comp, D', D'comp, S, S', and so on, every closed and bounded set is compact. Moreover, just as the (bounded) linear operators Rm→ Rn are elements of Rm ×Rn (i.e., n×m matrices) so too the bounded linear operators, say (where M1 and M2 are two smooth manifolds) are in oneto-one correspondence with distributions K (x,y) in the product manifold M1 ×M2. Half a century after its introduction, the Volterra integral was accordingly resuscitated:

using the physicists’ integral notation for the duality bracket. Under suitable hypotheses on supports and regularity with respect to x or y, this gives also a rigorous meaning to Volterra composition: the kernel of the composite K1&x K2 is the integral ∫K1(x,y)K2(y,z)dy. The analogy with the finite dimensional situation extends to the fact that this property is equivalent to the isomorphism of D'(M1 ×M2) with the tensor product with the hat indicating completion (in the sense of every “natural” topology on the tensor product): distributions ut(x,y) in M1 ×M2 are equal to “infinite” sums.
In all this, D’ can be replaced by any one of the other functional spaces above. Grothendieck took the Schwartz kernel theorem as the starting point for his theory of nuclear spaces (the kernel theorem is true because the spaces under consideration are nuclear), and the theorem now provides a safe base for the study of such special classes of operators as, for example, pseudodifferential operators or Fourier integral operators. One starts by analyzing the corresponding kernel distributions, and there is no real need to know the (relatively simple) proof of the Schwartz kernel theorem.

Distributions can be, and often are, taught at the undergraduate level, without any recourse to functional analysis, using solely sequences of test functions or of distributions. The success of distributions lies in the simplicity of the rules for handling them. It is what mathematical analysis needed at the midpoint of the twentieth century: an easy algorithm allowing the basic operations of, for example, differentiating or integrating under the integral sign, to be carried out without a second thought. With the easy part taken care of, analysts were then free to dig deeper and work on the finer and truly difficult problems. Beyond analysis, distributions provided the language for vast tracks of mathematics, pure and applied. Only three examples will be mentioned here.

In the 1950s and 1960s, François Bruhat and Harish-Chandra showed the role of distributions in noncommutative harmonic analysis. Harish-Chandra defined the natural generalization of the Schwartz space S for semi-simple Lie groups and put it to spectacular use in their representation theory.

In the 1940s the theorems of Georges De Rham on Hodge theory indicated that there was a duality between (singular) homology and (De Rham) cohomology on a C∞ manifold. However, this duality could not be formalized in mathematically acceptable analytic terms until the early 1950s, when De Rham learned of Schwartz’s theory of distributions and immediately introduced the necessary concept of currents, which are differential forms with distributional coefficients. The duality of currents with smooth differential forms having compact support is built into their definition, so the concept of currents can be extended to the exterior derivative, to Riemannian manifolds, and to the Hodge operations. In this way, co-cycles and co-boundaries become analytic objects: closed and
exact currents, respectively. Chains in singular homology, normally regarded as concrete geometric objects, can also be viewed as currents. During the 1960s and 1970s, what are currents of weak regularity have turned out to be important in the study of minimal surfaces and of analytic varieties.

In general, physicists turned out to have been better prepared than mathematicians to accept distributions, perhaps because of their acquaintance with the work of Dirac. The Heisenberg uncertainty principle had accustomed them to the fact that an observable could not, in general, be evaluated at a point but had to be tested over extended regions. As it turned out, at a deeper level the axiomatic theory of quantum electrodynamics needed a highly sophisticated version of distributions, the distributions with values in the set of unbounded linear operators on a Hilbert space, and this led to serious difficulties, by no means entirely resolved and which are rooted in the impossibility of multiplying arbitrary distributions. Attempts to circumvent these difficulties were made through renormalization.

Probability Theory At the end of the 1960s, Schwartz redirected his research towards infinite-dimensional measure and probability theory. His approach was through cylindrical Radon probabilities. A cylindrical Radon measure on, say, a separable Banach space E is the assignment to any pair (X, ft ) consisting of a finite-dimensional vector space X and of a continuous linear map f: E →X , of a Radon measure μf with the obvious coherence requirements: if Y is another finite-dimensional vector space, and if g : X→Y is linear, thenμg f = g(μ,f) A natural question is whether there is a Radon measure μ on E such that μf = (f, μ) for every pair (X, f ). Prokhorov’s theorem provides the necessary and sufficient condition for this to be true. Schwartz focused his attention on the linear maps u of E into another Banach space F that transform an important class of cylindrical measures on E into true Radon measures on F , which led him to the concept of p-Radonifying maps (p> 0). He proved that for p> 1 they are the same as the p-summing maps studied by other mathematicians. The map u is p-summing if it transforms every sequence {xn}n=1,2,.; such that
>an element of the dual E* of E )
, into a sequence such that. The same equivalence can be proved for 0 < p ≤ 1 under some slightly restrictive hypotheses on E or F . There is also an important Schwartz duality theorem: If E* can be embedded into an Lp-space, then the adjoint u*: F* → E* of an arbitrary p-summable map is p-summable. (Later, the embeddability of E * into Lp was proved to be necessary.)

In later work Laurent Schwartz turned his attention to random processes and proposed to define semimartingales {Xt} valued in a smooth (real or complex) manifold M by the property that (X) be a real or complex semi-martingale for every smooth map : M →R or C (here “smooth” means C2 or holomorphic). If M is complex but not Stein, for instance if M is compact, there might not be (enough) holomorphic functions for this definition to make sense, which led Schwartz to localize his definition of M-valued semimartingales. Following on his lifelong habit of exploiting duality, he gave a weak definition of dX through stochastic integrals ∫HdX in which H plays the role of a test-function. With this approach Schwartz was able to give a new, coordinate-free interpretation of the classical Ito formula as the equation d(f(Xt)=)d(X), where i dX s a special second-order stochastic differential operator on M.

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